Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Percentage Accurate: 85.3% → 98.0%

Time: 7.3s

Alternatives: 12

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{z - a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * (z - t)) / (z - a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}

Python

def code(x, y, z, t, a):
	return x + ((y * (z - t)) / (z - a))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y * (z - t)) / (z - a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{z - a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * (z - t)) / (z - a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}

Python

def code(x, y, z, t, a):
	return x + ((y * (z - t)) / (z - a))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(z - a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y * (z - t)) / (z - a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{t - z}{a - z}, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ (- t z) (- a z)) y x))

C

double code(double x, double y, double z, double t, double a) {
	return fma(((t - z) / (a - z)), y, x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(Float64(t - z) / Float64(a - z)), y, x)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(t - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]

Derivation

1. Initial program 88.2%

   \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   8. lower-/.f64 98.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

5. Final simplification 98.8%

   \[\leadsto \mathsf{fma}\left(\frac{t - z}{a - z}, y, x\right) \]
6. Add Preprocessing

Alternative 2: 83.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - z} \cdot \left(t - z\right)\\ t_2 := \frac{\left(t - z\right) \cdot y}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- a z)) (- t z))) (t_2 (/ (* (- t z) y) (- a z))))
   (if (<= t_2 -1e+24) t_1 (if (<= t_2 2e+74) (fma (/ z (- z a)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (a - z)) * (t - z);
	double t_2 = ((t - z) * y) / (a - z);
	double tmp;
	if (t_2 <= -1e+24) {
		tmp = t_1;
	} else if (t_2 <= 2e+74) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(a - z)) * Float64(t - z))
	t_2 = Float64(Float64(Float64(t - z) * y) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= -1e+24)
		tmp = t_1;
	elseif (t_2 <= 2e+74)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+24], t$95$1, If[LessEqual[t$95$2, 2e+74], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -9.9999999999999998e23 or 1.9999999999999999e74 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 72.8%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z - t\right) \cdot y}}{z - a} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \frac{y}{z - a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \frac{y}{z - a}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(z - t\right)} \cdot \frac{y}{z - a} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(z - t\right) \cdot \color{blue}{\frac{y}{z - a}} \]

      6. lower--.f64 83.6

         \[\leadsto \left(z - t\right) \cdot \frac{y}{\color{blue}{z - a}} \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(z - t\right) \cdot \frac{y}{z - a}} \]

   if -9.9999999999999998e23 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1.9999999999999999e74

   1. Initial program 100.0%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{z - a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{z - a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{z - a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{z - a} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{z - a}}, y, x\right) \]

      6. lower--.f64 95.8

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{z - a}}, y, x\right) \]

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - z\right) \cdot y}{a - z} \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{a - z} \cdot \left(t - z\right)\\ \mathbf{elif}\;\frac{\left(t - z\right) \cdot y}{a - z} \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot \left(t - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-12)
   (+ x y)
   (if (<= z -1.85e-77)
     (fma (/ (- t) z) y x)
     (if (<= z 7e+41) (fma (- t z) (/ y a) x) (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-12) {
		tmp = x + y;
	} else if (z <= -1.85e-77) {
		tmp = fma((-t / z), y, x);
	} else if (z <= 7e+41) {
		tmp = fma((t - z), (y / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-12)
		tmp = Float64(x + y);
	elseif (z <= -1.85e-77)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (z <= 7e+41)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-12], N[(x + y), $MachinePrecision], If[LessEqual[z, -1.85e-77], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 7e+41], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999997e-12 or 6.9999999999999998e41 < z

   1. Initial program 77.5%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 83.0

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 83.0%

      \[\leadsto \color{blue}{y + x} \]

   if -4.9999999999999997e-12 < z < -1.84999999999999998e-77

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

      8. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]

      2. lower--.f64 80.4

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - t}}{z}, y, x\right) \]

   7. Applied rewrites 80.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot t}{z}, y, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.1%

       \[\leadsto \mathsf{fma}\left(\frac{-t}{z}, y, x\right) \]

   if -1.84999999999999998e-77 < z < 6.9999999999999998e41

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{a}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{z - t}{a}}\right)\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{z - t}{a} \cdot y}\right)\right) + x \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a}\right)\right) \cdot y} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{z - t}{a}\right), y, x\right)} \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{a}}, y, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(z - t\right)}}{a}, y, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(z - t\right)}{a}}, y, x\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}{a}, y, x\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \left(z + \color{blue}{-1 \cdot t}\right)}{a}, y, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \color{blue}{\left(-1 \cdot t + z\right)}}{a}, y, x\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(-1 \cdot t\right) + -1 \cdot z}}{a}, y, x\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \left(-1 \cdot t\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{a}, y, x\right) \]

      15. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(-1 \cdot t\right) - z}}{a}, y, x\right) \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot t\right)\right)} - z}{a}, y, x\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right) - z}{a}, y, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t} - z}{a}, y, x\right) \]

      19. lower--.f64 84.6

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{a}, y, x\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.3%

      \[\leadsto \mathsf{fma}\left(t - z, \color{blue}{\frac{y}{a}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-12)
   (+ x y)
   (if (<= z -1.85e-77)
     (fma (/ (- t) z) y x)
     (if (<= z 5.2e-15) (fma (/ y a) t x) (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-12) {
		tmp = x + y;
	} else if (z <= -1.85e-77) {
		tmp = fma((-t / z), y, x);
	} else if (z <= 5.2e-15) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-12)
		tmp = Float64(x + y);
	elseif (z <= -1.85e-77)
		tmp = fma(Float64(Float64(-t) / z), y, x);
	elseif (z <= 5.2e-15)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-12], N[(x + y), $MachinePrecision], If[LessEqual[z, -1.85e-77], N[(N[((-t) / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 5.2e-15], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999997e-12 or 5.20000000000000009e-15 < z

   1. Initial program 78.8%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 81.7

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{y + x} \]

   if -4.9999999999999997e-12 < z < -1.84999999999999998e-77

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

      8. lower-/.f64 99.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]

      2. lower--.f64 80.4

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - t}}{z}, y, x\right) \]

   7. Applied rewrites 80.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot t}{z}, y, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.1%

       \[\leadsto \mathsf{fma}\left(\frac{-t}{z}, y, x\right) \]

   if -1.84999999999999998e-77 < z < 5.20000000000000009e-15

   1. Initial program 96.5%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot y}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot t} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]

      5. lower-/.f64 84.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t, x\right) \]

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-50}:\\ \;\;\;\;\frac{z - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-12)
   (+ x y)
   (if (<= z -4e-50)
     (* (/ (- z t) z) y)
     (if (<= z 5.2e-15) (fma (/ y a) t x) (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-12) {
		tmp = x + y;
	} else if (z <= -4e-50) {
		tmp = ((z - t) / z) * y;
	} else if (z <= 5.2e-15) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-12)
		tmp = Float64(x + y);
	elseif (z <= -4e-50)
		tmp = Float64(Float64(Float64(z - t) / z) * y);
	elseif (z <= 5.2e-15)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-12], N[(x + y), $MachinePrecision], If[LessEqual[z, -4e-50], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 5.2e-15], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999992e-12 or 5.20000000000000009e-15 < z

   1. Initial program 78.8%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 81.7

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{y + x} \]

   if -3.99999999999999992e-12 < z < -4.00000000000000003e-50

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

      8. lower-/.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z - t\right) \cdot y}}{z - a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(z - t\right) \cdot y}}{z - a} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(z - t\right)} \cdot y}{z - a} \]

      5. lower--.f64 83.9

         \[\leadsto \frac{\left(z - t\right) \cdot y}{\color{blue}{z - a}} \]

   7. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\frac{\left(z - t\right) \cdot y}{z - a}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.9%

       \[\leadsto \frac{z - t}{z} \cdot \color{blue}{y} \]

   if -4.00000000000000003e-50 < z < 5.20000000000000009e-15

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot y}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot t} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]

      5. lower-/.f64 82.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t, x\right) \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-12}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-50}:\\ \;\;\;\;\frac{z - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) z) y x)))
   (if (<= z -3.1e-53) t_1 (if (<= z 1.4e+41) (fma (- t z) (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / z), y, x);
	double tmp;
	if (z <= -3.1e-53) {
		tmp = t_1;
	} else if (z <= 1.4e+41) {
		tmp = fma((t - z), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / z), y, x)
	tmp = 0.0
	if (z <= -3.1e-53)
		tmp = t_1;
	elseif (z <= 1.4e+41)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -3.1e-53], t$95$1, If[LessEqual[z, 1.4e+41], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000015e-53 or 1.4e41 < z

   1. Initial program 79.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z}}, y, x\right) \]

      6. lower--.f64 88.1

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - t}}{z}, y, x\right) \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)} \]

   if -3.10000000000000015e-53 < z < 1.4e41

   1. Initial program 96.9%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{a}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{z - t}{a}}\right)\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{z - t}{a} \cdot y}\right)\right) + x \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a}\right)\right) \cdot y} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{z - t}{a}\right), y, x\right)} \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{a}}, y, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(z - t\right)}}{a}, y, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(z - t\right)}{a}}, y, x\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}{a}, y, x\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \left(z + \color{blue}{-1 \cdot t}\right)}{a}, y, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \color{blue}{\left(-1 \cdot t + z\right)}}{a}, y, x\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(-1 \cdot t\right) + -1 \cdot z}}{a}, y, x\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \left(-1 \cdot t\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{a}, y, x\right) \]

      15. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(-1 \cdot t\right) - z}}{a}, y, x\right) \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot t\right)\right)} - z}{a}, y, x\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right) - z}{a}, y, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t} - z}{a}, y, x\right) \]

      19. lower--.f64 84.0

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{a}, y, x\right) \]

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(t - z, \color{blue}{\frac{y}{a}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- z a)) y x)))
   (if (<= z -3.3e-51) t_1 (if (<= z 7e-21) (fma (- t z) (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / (z - a)), y, x);
	double tmp;
	if (z <= -3.3e-51) {
		tmp = t_1;
	} else if (z <= 7e-21) {
		tmp = fma((t - z), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(z - a)), y, x)
	tmp = 0.0
	if (z <= -3.3e-51)
		tmp = t_1;
	elseif (z <= 7e-21)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -3.3e-51], t$95$1, If[LessEqual[z, 7e-21], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.29999999999999973e-51 or 7.0000000000000007e-21 < z

   1. Initial program 80.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot z}{z - a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{z - a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{z - a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{z - a} \cdot y} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{z - a}}, y, x\right) \]

      6. lower--.f64 86.1

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{z - a}}, y, x\right) \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)} \]

   if -3.29999999999999973e-51 < z < 7.0000000000000007e-21

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{a}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{z - t}{a}}\right)\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{z - t}{a} \cdot y}\right)\right) + x \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a}\right)\right) \cdot y} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{z - t}{a}\right), y, x\right)} \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{a}}, y, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(z - t\right)}}{a}, y, x\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(z - t\right)}{a}}, y, x\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(t\right)\right)\right)}}{a}, y, x\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \left(z + \color{blue}{-1 \cdot t}\right)}{a}, y, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \color{blue}{\left(-1 \cdot t + z\right)}}{a}, y, x\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(-1 \cdot t\right) + -1 \cdot z}}{a}, y, x\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot \left(-1 \cdot t\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{a}, y, x\right) \]

      15. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1 \cdot \left(-1 \cdot t\right) - z}}{a}, y, x\right) \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot t\right)\right)} - z}{a}, y, x\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right) - z}{a}, y, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t} - z}{a}, y, x\right) \]

      19. lower--.f64 84.7

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{a}, y, x\right) \]

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.4%

      \[\leadsto \mathsf{fma}\left(t - z, \color{blue}{\frac{y}{a}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-50}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e-50) (+ x y) (if (<= z 5.2e-15) (fma (/ y a) t x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e-50) {
		tmp = x + y;
	} else if (z <= 5.2e-15) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e-50)
		tmp = Float64(x + y);
	elseif (z <= 5.2e-15)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e-50], N[(x + y), $MachinePrecision], If[LessEqual[z, 5.2e-15], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000007e-50 or 5.20000000000000009e-15 < z

   1. Initial program 80.6%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 77.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{y + x} \]

   if -1.70000000000000007e-50 < z < 5.20000000000000009e-15

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot y}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot t} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]

      5. lower-/.f64 82.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t, x\right) \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-50}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e-183) (+ x y) (if (<= z 3.3e-305) (/ (* y t) a) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-183) {
		tmp = x + y;
	} else if (z <= 3.3e-305) {
		tmp = (y * t) / a;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.1d-183)) then
        tmp = x + y
    else if (z <= 3.3d-305) then
        tmp = (y * t) / a
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-183) {
		tmp = x + y;
	} else if (z <= 3.3e-305) {
		tmp = (y * t) / a;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e-183:
		tmp = x + y
	elif z <= 3.3e-305:
		tmp = (y * t) / a
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e-183)
		tmp = Float64(x + y);
	elseif (z <= 3.3e-305)
		tmp = Float64(Float64(y * t) / a);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.1e-183)
		tmp = x + y;
	elseif (z <= 3.3e-305)
		tmp = (y * t) / a;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e-183], N[(x + y), $MachinePrecision], If[LessEqual[z, 3.3e-305], N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e-183 or 3.29999999999999982e-305 < z

   1. Initial program 86.4%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 66.9

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{y + x} \]

   if -2.1000000000000002e-183 < z < 3.29999999999999982e-305

   1. Initial program 99.9%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

      8. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot y}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

      4. lower-/.f64 94.0

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

   7. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e-183) (+ x y) (if (<= z 3.3e-305) (* (/ y a) t) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-183) {
		tmp = x + y;
	} else if (z <= 3.3e-305) {
		tmp = (y / a) * t;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.1d-183)) then
        tmp = x + y
    else if (z <= 3.3d-305) then
        tmp = (y / a) * t
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-183) {
		tmp = x + y;
	} else if (z <= 3.3e-305) {
		tmp = (y / a) * t;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e-183:
		tmp = x + y
	elif z <= 3.3e-305:
		tmp = (y / a) * t
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e-183)
		tmp = Float64(x + y);
	elseif (z <= 3.3e-305)
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.1e-183)
		tmp = x + y;
	elseif (z <= 3.3e-305)
		tmp = (y / a) * t;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e-183], N[(x + y), $MachinePrecision], If[LessEqual[z, 3.3e-305], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e-183 or 3.29999999999999982e-305 < z

   1. Initial program 86.4%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 66.9

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{y + x} \]

   if -2.1000000000000002e-183 < z < 3.29999999999999982e-305

   1. Initial program 99.9%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

      8. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot y}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

      4. lower-/.f64 94.0

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

   7. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   12. Step-by-step derivation

   13. Applied rewrites 62.2%

       \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e-183) (+ x y) (if (<= z 3.3e-305) (* (/ t a) y) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-183) {
		tmp = x + y;
	} else if (z <= 3.3e-305) {
		tmp = (t / a) * y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.1d-183)) then
        tmp = x + y
    else if (z <= 3.3d-305) then
        tmp = (t / a) * y
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-183) {
		tmp = x + y;
	} else if (z <= 3.3e-305) {
		tmp = (t / a) * y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e-183:
		tmp = x + y
	elif z <= 3.3e-305:
		tmp = (t / a) * y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e-183)
		tmp = Float64(x + y);
	elseif (z <= 3.3e-305)
		tmp = Float64(Float64(t / a) * y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.1e-183)
		tmp = x + y;
	elseif (z <= 3.3e-305)
		tmp = (t / a) * y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e-183], N[(x + y), $MachinePrecision], If[LessEqual[z, 3.3e-305], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e-183 or 3.29999999999999982e-305 < z

   1. Initial program 86.4%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 66.9

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{y + x} \]

   if -2.1000000000000002e-183 < z < 3.29999999999999982e-305

   1. Initial program 99.9%

      \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(z - t\right)}}{z - a} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

      8. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{z - a}}, y, x\right) \]

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{t \cdot y}{a}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot y}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

      4. lower-/.f64 94.0

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

   7. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.5%

       \[\leadsto \frac{t}{a} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x y))

C

double code(double x, double y, double z, double t, double a) {
	return x + y;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + y
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + y;
}

Python

def code(x, y, z, t, a):
	return x + y

Julia

function code(x, y, z, t, a)
	return Float64(x + y)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + y;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 88.2%

   \[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{y + x} \]

   2. lower-+.f64 60.6

      \[\leadsto \color{blue}{y + x} \]

5. Applied rewrites 60.6%

   \[\leadsto \color{blue}{y + x} \]

6. Final simplification 60.6%

   \[\leadsto x + y \]
7. Add Preprocessing

Developer Target 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ y (/ (- z a) (- z t)))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - t)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y / ((z - a) / (z - t)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - t)));
}

Python

def code(x, y, z, t, a):
	return x + (y / ((z - a) / (z - t)))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(y / Float64(Float64(z - a) / Float64(z - t))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (y / ((z - a) / (z - t)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

  (+ x (/ (* y (- z t)) (- z a))))